INTEGRATION OF DIFFERENTIAL EQUATIONS. 105 



may be reduced to 



d*y 1 dy > 

 S-SaS + rt' 88 ' ' 



for the coefficient of a n in the former is 



n (n - 2) - jp (p - 2) = (w -^) (n +^ - 2), 



which, provided p is even, may be reduced to n (n 2). But in 

 this, and in analogous cases, the auxiliary equation is, appa- 

 rently, insoluble. 



The applicability of our transformation would, it is evident, 

 not be affected, if the equation were, instead of (25), 



and, provided p or p 1 were divisible by m t, (29) might be 

 reduced to 



But this case requires more care than those already con- 

 sidered, as if certain factors which apparently disappear are 

 neglected, our solution is incomplete, or erroneous. 



An instance will make this clear, 



dy y 



Here (n - 2) (n 4- 1) a n + (n - l) q a^ = .. ........ (A). 



Let (n + 1) a n = (n - 1) b n .................. (a), 



... ( w -2)(-l)n5 n +(-2)(-l)^ 1 = ...... (B), 



The factor n 2 may be safely neglected. But n 1 is 

 essential, because the solution of the auxiliary equation 



dz 



gives (n - 1) (nl n + qb^ = 0, 



and would be incomplete if we omitted the first factor. 

 From (a) we get 



